Vassiliki is from the Greek Βασιλική (Vassilikí — queenly, royal) — alternative transliteration of Vasiliki. Vassiliki is one of the Greek feminine names — central to traditional Greek Orthodox heritage. Peter's Basilica Vatican Royal Stoa Athens.
Vassiliki does not currently appear in the US Social Security Administration's top 1,000 girls' names, so we don't publish a US rank or birth count for it. That says nothing about the name's standing elsewhere in the world — only that it sits outside the ranked US data we rely on.
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In Pythagorean numerology the letters of Vassiliki reduce to 3, The Communicator. This is a traditional interpretive system, not a factual claim about the name.